𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications

Baazeem, Amani; Dragomir, Silvestru Sever; Feki, Kais

doi:10.3390/math13213393

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𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications

by

Amani Baazeem

¹,

Silvestru Sever Dragomir

²

and

Kais Feki

^3,*

¹

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia

²

Applied Mathematics Research Group, Victoria University, P.O. Box 14428, Melbourne, MC 8001, Australia

³

Department of Mathematics, College of Sciences and Arts, Najran University, P.O. Box 1988, Najran 11001, Saudi Arabia

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(21), 3393; https://doi.org/10.3390/math13213393 (registering DOI)

Submission received: 23 August 2025 / Revised: 9 October 2025 / Accepted: 18 October 2025 / Published: 24 October 2025

(This article belongs to the Section C: Mathematical Analysis)

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Abstract

Consider a complex Hilbert space

(X, ⟨ \cdot, \cdot ⟩)

equipped with a positive (semidefinite) bounded linear operator

A

on

X

. The

A

-joint numerical radius for two

A

-bounded operators

T

and

S

is defined as

ω_{A, e} (T, S) = {sup}_{{∥ x ∥}_{A} = 1} \sqrt{{|{⟨ T x, x ⟩}_{A}|}^{2} + {|{⟨ S x, x ⟩}_{A}|}^{2}} .

Among other results, in this study we demonstrate that

ω_{e, A}^{2} (T, S) \leq 2^{1 - \frac{1}{r}} {[max \{{∥T∥}_{A}^{2 r}, {∥S∥}_{A}^{2 r}\} + ω_{A}^{r} (S^{♯_{A}} T)]}^{\frac{1}{r}}

for

r \geq 1

and that

ω_{e, A}^{2} (T, S) \geq \frac{1}{2} ω_{A} (T^{2} + S^{2}) + \frac{1}{2} max \{ω_{A} (S + T), ω_{A} (S - T)\} |ω_{A} (S) - ω_{A} (T)|

. Additionally, we provide several inequalities related to the

A

-numerical radius and

A

-seminorm.

Keywords: joint ?-numerical radius; Euclidean A-seminorm; semi-inner product; positive operator

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MDPI and ACS Style

Baazeem, A.; Dragomir, S.S.; Feki, K. 𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications. Mathematics 2025, 13, 3393. https://doi.org/10.3390/math13213393

AMA Style

Baazeem A, Dragomir SS, Feki K. 𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications. Mathematics. 2025; 13(21):3393. https://doi.org/10.3390/math13213393

Chicago/Turabian Style

Baazeem, Amani, Silvestru Sever Dragomir, and Kais Feki. 2025. "𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications" Mathematics 13, no. 21: 3393. https://doi.org/10.3390/math13213393

APA Style

Baazeem, A., Dragomir, S. S., & Feki, K. (2025). 𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications. Mathematics, 13(21), 3393. https://doi.org/10.3390/math13213393

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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